viete-like_formulas.sf

#!/usr/bin/ruby

#
## Viète-like formulas:
#

# a(c) = {
#    f(0) = 0
#    f(n) = sqrt(c + f(n-1))
#    d = (1 + sqrt(1 + 4*c))/2
#    Limit_{n -> Infinity} (2*d)^n * d - (2*d)^n * f(n)
# }

# Closed-forms:
#   a(1) = 2*P             where P is the Paris constant (https://oeis.org/A105415)
#   a(2) = pi^2/4

# See also:
#   https://en.wikipedia.org/wiki/Vi%C3%A8te%27s_formula

local Number!PREC = 1024.numify
const n = 100

func a(c) {

    func f(n) {
        n == 0 ? 0 : sqrt(c + f(n-1))
    }

    var d = (1 + sqrt(1 + 4*c))/2
    var t = ((2*d)**n * d - ((2*d)**n * f(n)))

    return t
}

for c in (1..10) {
    say ("a(#{c}) = ", a(c).as_dec(48))
}

__END__
a(1) = 2.19728392878831297146933783468719242174669679222
a(2) = 2.46740110027233965470862274996903778382842485181
a(3) = 2.72432397395262808042738254231361436370470931728
a(4) = 2.95697498038250320240181341327815809572441049058
a(5) = 3.16942068660729603631214225411774841781355055152
a(6) = 3.36565753974384094582778152437432384197290834457
a(7) = 3.54871470954831491531526350481479738940975906022
a(8) = 3.72084501177716618448006573526170234977853589736
a(9) = 3.88375012406073194374765723289729464351431116217
a(10) = 4.03874313819098595611125546305106699051058830246